In the diagram above, the left triangle commutes and the straight line above is nuclear.
Since a composition of a bounded linear map and a nuclear map is again nuclear,
we only have to show that \(\tau _k^{-1}\)  is bounded.
For this purpose, let us recall how explicitly \(\lbrace \widetilde{x}_{l,m}\rbrace \)  are obtained
(see [1]} for a corresponding formula for \(S^1\) -action):
\(\widetilde{x}_{l,m} = \frac{1}{N^2}\sum _{j_1,j_2} e^{-i\frac{2\pi (j_1l +j_2m)}{N}} {\hbox{\rm Ad\,}}(V_{\mathrm {c}}^{j_1}\otimes V_{\mathrm {c}}^{j_2})(\widetilde{x}).\) 
